Robust topology optimization of hinge-free compliant mechanisms with material uncertainties based on a non-probabilistic field model

Abstract

This paper presents a new robust topology optimization framework for hinge-free compliant mecha- nisms with spatially varying material uncertainties, which are described using a non-probabilistic bounded field model. Bounded field uncertainties are efficiently represented by a reduced set of uncertain-but-bounded coefficients on the basis of the series expansion method. Robust topology optimization of compliant mechanisms is then defined to minimize the variation in output displacement under constraints of the mean displacement and predefined material volume. The nest optimization problem is solved using a gradient-based optimization algorithm. Numerical examples are presented to illustrate the effectiveness of the proposed method for circumventing hinges in topology optimization of compliant mechanisms.

Keywords

compliant mechanisms / robust topology optimization / hinges / uncertainty / bounded field

References

[1] Shi F, Ramesh P, Mukherjee S. Simulation methods for micro-electro-mechanical structures (MEMS) with application to a microtweezer. Computers & Structures, 1995, 56(5): 769–783

[2] Kota S, Joo J, Li Z, Design of compliant mechanisms: Applications to MEMS. Analog Integrated Circuits and Signal Processing, 2001, 29(1‒2): 7–15

[3] Sigmund O, Maute K. Topology optimization approaches. Structural and Multidisciplinary Optimization, 2013, 48(6): 1031–1055

[4] Deaton J D, Grandhi R V. A survey of structural and multidisciplinary continuum topology optimization: Post 2000. Structural and Multidisciplinary Optimization, 2014, 49(1): 1–38

[5] Pedersen C B W, Buhl T, Sigmund O. Topology synthesis of large-displacement compliant mechanisms. International Journal for Numerical Methods in Engineering, 2001, 50(12): 2683–2705

[6] Lee E, Gea H C. A strain based topology optimization method for compliant mechanism design. Structural and Multidisciplinary Optimization, 2014, 49(2): 199–207

[7] Ansola R, Veguería E, Canales J, A simple evolutionary topology optimization procedure for compliant mechanism design. Finite Elements in Analysis and Design, 2007, 44(1–2): 53–62

[8] Huang X, Li Y, Zhou S W, et al. Topology optimization of compliant mechanisms with desired structural stiffness. Engineering Structures, 2014, 79: 13–21

[9] Luo Z, Tong L. A level set method for shape and topology optimization of large-displacement compliant mechanisms. International Journal for Numerical Methods in Engineering, 2008, 76(6): 862–892

[10] Zhu B, Zhang X. A new level set method for topology optimization of distributed compliant mechanism. International Journal for Numerical Methods in Engineering, 2012, 91(8): 843–871

[11] Luo Z, Zhang N, Ji J, A meshfree level-set method for topological shape optimization of compliant multiphysics actuators. Computer Methods in Applied Mechanics and Engineering, 2012, 223–224: 133–152

[12] Saxena A, Ananthasuresh G K. On an optimal property of compliant topologies. Structural and Multidisciplinary Optimization, 2000, 19(1): 36–49

[13] Hetrick J A, Kota S. An energy formulation for parametric size and shape optimization of compliant mechanisms. Journal of Mechanical Design, 1999, 121(2): 229–234

[14] Sigmund O. On the design of compliant mechanisms using topology optimization. Mechanics Based Design of Structures and Machines, 1997, 25(4): 493–524

[15] Deepak S R, Dinesh M, Sahu D K, A comparative study of the formulations and benchmark problems for the topology optimization of compliant mechanisms. Journal of Mechanisms and Robotics, 2009, 1(1): 011003

[16] Poulsen T A. A simple scheme to prevent checkerboard patterns and one-node connected hinges in topology optimization. Structural and Multidisciplinary Optimization, 2002, 24(5): 396–399

[17] Wang N, Zhang X. Compliant mechanisms design based on pairs of curves. Science China. Technological Sciences, 2012, 55(8): 2099–2106

[18] Luo Z, Chen L, Yang J, Compliant mechanisms design using multi-objective topology optimization scheme of continuum structures. Structural and Multidisciplinary Optimization, 2005, 30(2): 142–154

[19] Zhu B, Zhang X, Wang N. Topology optimization of hinge-free compliant mechanisms with multiple outputs using level set method. Structural and Multidisciplinary Optimization, 2013, 47(5): 659–672

[20] Zhu B, Zhang X, Fatikow S. A multi-objective method of hinge-free compliant mechanism optimization. Structural and Multidisciplinary Optimization, 2014, 49(3): 431–440

[21] Lopes C G, Novotny A A. Topology design of compliant mechanisms with stress constraints based on the topological derivative concept. Structural and Multidisciplinary Optimization, 2016, 54(4): 737–746

[22] de Assis Pereira A, Cardoso E L. On the influence of local and global stress constraint and filtering radius on the design of hinge-free compliant mechanisms. Structural and Multidisciplinary Optimization, 2018, 58(2): 641–655

[23] Luo Y, Kang Z, Luo Z, Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Structural and Multidisciplinary Optimization, 2009, 39(3): 297–310

[24] Chen S, Chen W, Lee S. Level set based robust shape and topology optimization under random field uncertainties. Structural and Multidisciplinary Optimization, 2010, 41(4): 507–524

[25] Luo Y, Zhou M, Wang M Y, Reliability based topology optimization for continuum structures with local failure constraints. Computers & Structures, 2014, 143: 73–84

[26] Maute K, Frangopol D M. Reliability-based design of MEMS mechanisms by topology optimization. Computers & Structures, 2003, 81(8–11): 813–824

[27] Lazarov B S, Schevenels M, Sigmund O. Topology optimization considering material and geometric uncertainties using stochastic collocation methods. Structural and Multidisciplinary Optimization, 2012, 46(4): 597–612

[28] Doltsinis I, Kang Z. Robust design of structures using optimization methods. Computer Methods in Applied Mechanics and Engineering, 2004, 193(23–26): 2221–2237

[29] Sandgren E, Cameron T M. Robust design optimization of structures through consideration of variation. Computers & Structures, 2002, 80(20–21): 1605–1613

[30] Asadpoure A, Tootkaboni M, Guest J K. Robust topology optimization of structures with uncertainties in stiffness—Application to truss structures. Computers & Structures, 2011, 89(11–12): 1131–1141

[31] Vanmarcke E. Random Fields: Analysis and Synthesis. Singapore: World Scientific Publishing, 2010

[32] Jiang C, Li W, Han X, Structural reliability analysis based on random distributions with interval parameters. Computers & Structures, 2011, 89(23–24): 2292–2302

[33] Do D M, Gao W, Song C, Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters. International Journal for Numerical Methods in Engineering, 2016, 108(10): 1198–1229

[34] Ying X, Lee S, Chen W, Efficient random field uncertainty propagation in design using multiscale analysis. Journal of Mechanical Design, 2009, 131(2): 021006

[35] Luo Y, Zhan J, Xing J, Non-probabilistic uncertainty quantification and response analysis of structures with a bounded field model. Computer Methods in Applied Mechanics and Engineering, 2019 (in press)

[36] Svanberg K. The method of moving asymptotes—A new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359–373

[37] Sigmund O. Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 2007, 33(4–5): 401–424

[38] Wang F, Lazarov B S, Sigmund O. On projection methods, convergence and robust formulations in topology optimization. Structural and Multidisciplinary Optimization, 2011, 43: 767– 784

Acknowledgements

This work was financially supported by the National Key R&D Program of China (Grant No. 2017YFB0203604) and the National Natural Science Foundation of China (Grant No. 11472215).

RIGHTS & PERMISSIONS

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature